On rescalability of generators of hyperdifferential operators

Abstract

Properties of a given function can be studied and discovered by looking at their different representations. Much of the progress of the study of special functions, which are often used in physics, are attributed by the diverse library of their representations and identities. Hence, uncovering new representations and identities of functions, in general, is indubitably important. In Ref. [1], polynomial sequences were expressed using hyperdifferential operators. In the same study, rescalability was first defined and was used to develop a calculus to close finite summations of polynomial sequences. The main idea is to express a function with hyperdifferential operators and if rescalability is present, then an identity can be uncovered. In this paper we take a closer look into the concept of rescalability. We see the potential of rescalability as a mathematical tool and we wish to develop a general theory of rescalability. To start, we established two theorems in this paper that gives us the generalized form of a rescalable generator and a condition that tells us which polynomials may never be rescalable.